Tomographic Reconstruction in the 21st Century

Image reconstruction from projections is the field that lays the foundations for computed tomography (CT). For several decades, the established principles were applied not only to medical scanners in radiology and nuclear medicine but also to industrial scanning. When speaking of image reconstruction from projections, one is generally considering the problem of recovering some density function from measurements taken over straight lines, or the "line-integral model" for short. Image reconstruction can be performed by directly applying analytic formulas derived from the theory or by using general optimization methods adapted to handling large linear systems. The latter techniques are referred to as iterative to distinguish them from the analytic (or direct) methods. This article considers only the analytic methods. The two-dimensional (2-D) reconstruction problem (or classical tomography) refers to a density function in two dimensions with measurement lines lying in the plane, and the three-dimensional (3-D) problem considers 3-D density functions and lines with arbitrary orientations in space. The widely used term 3-D imaging is potentially confusing in this context, because there are some 3-D forms of image reconstruction that are mathematically equivalent to performing 2-D reconstruction on a set of parallel contiguous planes. To emphasize the distinction between 2-D and 3-D reconstruction, the terminology fully 3-D image reconstruction (or sometimes truly 3-D reconstruction) was introduced in the late 1980s when there seemed to be very little left to do in two dimensions but a rich unexplored 3-D theory to be developed.